## One Explanation For Friday the 13th’s Chance

So to give one answer to my calendar puzzle, which you may recall as this: for any given month and year, we know with certainty whether there’s a Friday the 13th in it. And yet, we can say that “Friday the 13ths are more likely than any other day of the week”, and mean something by it, and even mean something true by it. Thanks to the patterns of the Gregorian calendar we are more likely to see a Friday the 13th than we are a Thursday the 13th, or Tuesday the 13th, or so on. (We’re also more likely to see a Saturday the 14th than the 14th being any other day of the week, but somehow that’s not so interesting.)

Here’s one way to look at it. In December 2011 there’s zero chance of encountering a Friday the 13th. As it happens, 2011 has only one month with a Friday the 13th in it, the lowest case which happens. In January 2012 there’s a probability of one of encountering a Friday the 13th; it’s right there on the schedule. There’ll also be Fridays the 13th in April and July of 2012. For the other months of 2012, there’s zero probability of encountering a Friday the 13th.

Imagine that I pick one of the months in either 2011 or 2012. What is the chance that it has a Friday the 13th? If I tell you which month it is, you know right away the chance is zero or one; or, at least, you can tell as soon as you find a calendar. Or you might work out from various formulas what day of the week the 13th of that month should be, but you’re more likely to find a calendar before you are to find that formula, much less work it out.

## How Did Friday The 13th Get A Chance?

Here’s a little puzzle in probability which, in a slightly different form, I gave to my students to work out. I get the papers back tomorrow. To brace myself against that I’m curious what my readers here would make of it.

Possibly you’ve encountered a bit of calendrical folklore which says that Friday the 13ths are more likely than any other day of the week’s 13th. That’s not that there are more Fridays the 13th than all the other days of the week combined, but rather that a Friday the 13th is more likely to happen than a Thursday the 13th, or a Sunday, or what have you. And this is true; one is slightly more likely to see a Friday the 13th than any other specific day of the week being that 13.

And yet … there’s a problem in talking about the probability of any month having a Friday the 13th. Arguably, no month has any probability of holding a Friday the 13th. Consider.

Is there a Friday the 13th this month? For the month of this writing, December 2011, the answer is no; the 13th is a Tuesday; the Fridays are the 2nd, 9th, 16th, 23rd, and 30th. But were this January 2012, the answer would be yes. For February 2012, the answer is no again, as the 13th comes on a Monday. But altogether, every month has a Friday the 13th or it hasn’t. Technically, we might say that a month which definitely has a Friday the 13th has a probability of 1, or 100%; and a month which definitely doesn’t has a probability of 0, or 0%, but we tend to think of those as chances in the same way we think of white or black as colors, mostly when we want to divert an argument into nitpicking over definitions.